%!TEX root = thesis.tex
\chapter{Conclusion}\label{chap:conclusion}

\section{Applications}

In \cite{BuhlerReichstein1997On-the-essentia} and \cite{BuhlerReichstein1999On-Tschirnhaus-}, the main application of essential dimension was to determine how much a ``general polynomial of degree $n$'' can be simplified via non-degenerate Tschirnhaus transformations.  It is shown in these papers that $\ed(S_n)$ is the minimal number of algebraically independent coefficients possible for a polynomial simplified in this manner.  This builds on classical results due to Klein, Hermite and Joubert.

Consider a general univariate polynomial equation of degree $n$.  One may find an expression for the solution in terms of the coefficients of the polynomial, the field operations, and compositions of some family of functions.  For example, for polynomials of degree $\le 4$, one can use the family of radical functions.  In a modern interpretation~\cite{Dixmier1993Hilbert}, Hilbert's thirteenth problem is to determine the minimal number of variables $s(n)$ required for any such family of functions (Hilbert was interested in $n=7$).  When one allows continuous functions, Arnol'd~\cite{Arnold1957} and Kolmogorov~\cite{Kolmogorov1957} showed that $s(n)=1$.  When one considers algebraic functions, the question is open.

The value of $\ed(S_n)$ is an upper bound for $s(n)$ and the best lower bound for $\ed(S_n)$ is much lower than the best upper bounds for $s(n)$.  Thus, any improvement on estimates for $s(n)$ or $\ed(S_n)$ would be a significant advance.  As Hilbert was interested in $n=7$, Theorem \ref{thm:edS7} can be interpreted as a solution to a variant of Hilbert's 13th problem.

More generally, consider a finite group $G$.  We outline the connections with essential dimension, Noether's problem, and generic polynomials.  The interested reader should see \cite{Saltman1982Generic_Galois_}, \cite{DeMeyer1983Generic_polynom}, \cite{KemperMattig2000Generic-polynom}, and \cite{JensenLedetYui2002Generic-polynom}.

\emph{Noether's problem} asks whether the field $k(V)^G$ is rational over $k$, where $V$ is the regular representation of $G$.  The original interest in this question was motivated by an attempt to construct field extensions of $\bbQ$ with Galois group $G$.  In modern language, a positive answer means there exists a generic polynomial for $G$ over $k$.  When $k$ is Hilbertian (for example, when $k=\bbQ$), a generic polynomial over $k$ yields a field extension over $k$ with Galois group $G$.

In fact, a generic polynomial for $G$ over $k$ exists when $k(V)^G$ is only \emph{retract rational} over $k$.  Retract rationality is a property of varieties weaker than rationality but stronger than unirationality (see \cite{Saltman1982Generic_Galois_}).  When a versal variety $X$ has a retract rational quotient $X/G$, one can construct a generic polynomial with $\dim(X)$ algebraically independent parameters.  In fact, when a versal variety has a retract rational quotient, so does any variety corresponding to a faithful linear representation.  Since unirational and rational coincide in dimension $2$, for every group $G$ in Theorem \ref{thm:ed2classification}, there exists a versal $G$-variety $X$ whose rational quotient $X/G$ is rational.  Thus, we have the following corollary:

\begin{cor}
Let $k$ be a field of characteristic $0$.  If $G$ is a finite group of generic dimension $\le 2$ over $k$ then $G$ is isomorphic one of the groups in the statement of Theorem \ref{thm:ed2classification}.
\end{cor}

\section{Future Research}

In both of the major results of this thesis, it was crucial to identify whether or not a $G$-variety was versal.  Thus, we have the following recognition problem:

\begin{ques}
Are there good criteria for determining whether a given $G$-variety is versal?
\end{ques}

Even in dimension $2$ over $\bbC$, the question is unanswered.  Theorem \ref{thm:ed2classification} does not classify all versal minimal rational $G$-surfaces; it only identifies which groups appear.   Indeed, different $G$-surfaces with the same group $G$ may not be equivariantly birationally equivalent.  For example, there exist two versal $S_5$-actions on the following surfaces which are not equivariantly birationally equivalent: the Clebsch diagonal cubic (versal by a result of Hermite, see \cite{Coray1987Cubic_hypersurf}, \cite{ReichsteinYoussin2002Conditions_sati} and \cite{Kraft2006A_result_of_Her}) and the del Pezzo surface of degree $5$ (versal by the proof of Theorem \ref{thm:4surfacesClassification}).  Other examples of this phenomenon can be found for abelian groups \cite{ReichsteinYoussin2002A-birational-in}, for versal actions of $S_4$ and $A_5$ \cite{BannaiTokunaga2007A-note-on-embed} and for $S_3 \times C_2$ \cite{Iskovskikh2003Two_nonconjugat}.

If we vary the base field, we do not even know the groups.  For algebraically closed fields of non-zero characteristic, the coarse Enriques-Manin-Iskovskikh classification still holds.  However, the Dolgachev-Iskovskikh classification no longer applies.  Furthermore, unirational surfaces are not necessarily rational in this case, so the classification may be inadequate.  See \cite{Serre2008Le-group-de-cre} for related discussion.

For non-algebraically closed fields of characteristic $0$, we know that any group of essential dimension $2$ must be in the list from Theorem \ref{thm:ed2classification} (this is immediate from \cite[Proposition 1.5]{BerhuyFavi2003Essential-dimen}).  However, the problem of determining which groups appear is more complicated.  It is possible that a versal $G$-surface over a field $k$ may not be defined over a subfield $k'$ while there may be another versal $G$-surface that \emph{is} defined over $k'$.  A full classification of versal minimal rational $G$-surfaces would remedy this situation.

In higher dimensions, the problem is significantly more difficult.  First, even over $\bbC$, there exist unirational varieties that are not rational.  Second, there is no analog of the Enriques-Manin-Iskovskikh classification here, nor the Dolgachev-Iskovskikh classification.  In fact, until Prokhorov's paper \cite{Prokhorov2009Simple-finite-s}, it was an open question as to whether \emph{all} finite groups could be embedded into the Cremona group of rank $3$ \cite[6.0]{Serre2009A-Minkowski-sty}.

Recently, Beauville \cite{Beauville2011On_finite_simpl} has generalised our Theorem \ref{thm:edS7} as follows:

\begin{thm}
If $G$ is a finite simple group of essential dimension $3$ then $G$ is isomorphic to $A_6$ or $\PSL_2(\bbF_{11})$.
\end{thm}

Note, however, that the question of whether or not $\PSL_2(\bbF_{11})$ actually has essential dimension $3$ is still open.

For toric varieties, the techniques of Chapter \ref{chap:toricVersal} should be quite effective for studying the versality condition.  It is unclear how these may be generalised to other varieties.  There does not seem to be any compelling reason why Corollary \ref{cor:versalByPGroups} should only be true for toric varieties since versality is a birational invariant.  One might conjecture that this theorem holds for \emph{any} variety:

\begin{conj}\label{conj:versalByPGroups}
Let $G$ be a finite group acting faithfully on a variety $X$.  Then $X$ is $G$-versal if and only if, for any prime $p$, $X$ is $G_p$-versal for a Sylow $p$-subgroup of $G$.
\end{conj}

We conclude by mentioning a promising result regarding the recognition problem for versality.  In this thesis, all versal varieties are constructed as the image of a compression from a linear $G$-variety.  Until very recently, this was essentially the only known construction for versal varieties.  However, in \cite[Proposition 3.3]{ColKunPopZin2009Is_the_function}, a new construction for homogeneous spaces is discussed.
